风险结构问题研究
|
摘要
风险理论是现代精算和数学界研究的热点,风险结构是风险理论的重要内容。在假设风险相互独立的情况下,经典风险理论主要处理保险事务中的随机风险模型,讨论有限时间内的生存概率以及最终破产概率问题。对随机风险模型,研究较多的是连续时间模型,且大都集中于复合Poisson过程的风险模型,而对离散时间模型则研究的较少。而作为连续时间的离散化,离散时间风险模型意义直观,在实践中更易于应用,关于离散时间风险模型,讨论最多的是复合二项风险模型,二项风险模型的特点是其理赔次数的均值大于其方差,特别适用于同质性保单组合的理赔次数模型,而当保单组合的理赔次数观察分布的样本方差大于其均值时,显然用复合二项风险模型不再合适、本文研究离散的复合负二项风险模型,利用鞅理论,在理赔次数的方差大于均值条件下,证明了该模型的最终破产概率,证明方法有较大的改进。本研究中,将每张保单的保费及单位时间内收取保费次数都推广为随机变量,建立了保费完全随机的复合负二项风险模型,使之更加贴近保险公司经营实际,并讨论该模型的性质,利用离散鞅理论得到它的最终破产概率及Lundberg不等式。
上述各种风险模型适用于单一的险种,且保费收入过程与理赔过程相互独立,现实情况是保险公司经营多种险种,各险种的风险不一定严格符合现有的风险模型,但还想了解各组合资产风险,此时经典风险理论不再适用。本文将利用风险价值VaR(Value at risk)对组合资产风险进行研究。当风险超出自身的承受范围,那么保险商就会考虑进行再保险。已有学者研究了单个资产的风险值、停止损失保费及满足一些特定关系的风险组合的风险值,本文研究了理赔损失分布在一般情况下封闭式集合风险模型的风险值及停止损失保费问题,给出了具体的计算方法。最后运用非负随机变量函数凸序研究随机资产的风险值(VaR),证明了任意资产组合的风险不会超过其各个资产风险值之和,给出了组合资产风险值的上界。
风险结构问题是风险理论研究的重要内容,对金融企业控制风险有重要参考意义。本文研究了更加符合保险公司经营现实的风险模型及资产组合风险的VaR,对风险结构及相关问题进行探讨,着重从理论上研究如何预防破产的发生,所获得研究结果为金融企业控制风险提供理论依据。
Risk theory is a hot spot in modern actuarial science and risk structure is an important content in risk theory.On the assumption that risk is mutually independent, Classical risk theories give priority to stochastic risk models of insurance and survival or ruin probability within a limited time.In stochastic risk models,we consider mostly continual time risk models,especially compound Poisson procession risk models, discrete risk models are seldom treated,but discrete risk models are easier to perceive and more applicable.When treating discrete risk models,we often mention a compound binomial risk model whose characteristic is that the mean of claim numbers is greater than the variance of claim numbers,a compound binomial risk model is suitable for homogeneous policies assemblages,has been discussed.Actually policies assemblages have more or less non-homogeneity,a compound negative binomial risk model is more suitable for non-homogeneous policies assemblages than a compound binomial risk model and the property is that the variance of claim numbers is greater than the mean of claim numbers.This thesis Studies a compound negative binomial risk model whose formula of ultimate ruin probability has been proved by applying discrete martingale theory,the method is different form others references.We construct a compound negative binomial risk model with a completely stochastic premium where the premium of every policy and the number of insure charges at per unit time are random variables.This paper discusses some properties of a compound negative binomial risk model with a completely stochastic premium.By applying discrete martingale theory,this article proves the formula of ultimate ruin probability and the Lundberg inequality.
When only considering a policy whose premium and claim are independent each other,we can apply all sorts of risk models.But every insurer has a lot of policies whose premium and claim don't accord with existing risk models,this results that we can not apply the classics risk theories and use value at risk(VaR)to study a portfolio. Insures reinsure a policy whose loss exceeds the insurer's anticipation.Some scholars have studied an asset's VaR and stop-loss premium and some special portfolio's VaR, this thesis has discussed arbitrary asset portfolio's VaR and stop-loss premium, analytical expressions for risk value and stop-loss premiums of sums of independent random variables have been given.At finial,function convex order of non—negative random variables is introduced for random assets' VaR,the conclusion is that portfolio assets VaR can not exceed the sum of individual asset VaR,and the upper of portfolio is also given.
Risk structure problem is an important aspect of risk theory and is important significance for controlling risk of financial enterprises.This thesis has studied a new risk model which is more in line with the reality of insurers and arbitrary assets' portfolio's VaR,we study risk structure problem to guard against the occurrence of bankruptcy and our results are the gist for controlling risk of financial enterprises.
上述各种风险模型适用于单一的险种,且保费收入过程与理赔过程相互独立,现实情况是保险公司经营多种险种,各险种的风险不一定严格符合现有的风险模型,但还想了解各组合资产风险,此时经典风险理论不再适用。本文将利用风险价值VaR(Value at risk)对组合资产风险进行研究。当风险超出自身的承受范围,那么保险商就会考虑进行再保险。已有学者研究了单个资产的风险值、停止损失保费及满足一些特定关系的风险组合的风险值,本文研究了理赔损失分布在一般情况下封闭式集合风险模型的风险值及停止损失保费问题,给出了具体的计算方法。最后运用非负随机变量函数凸序研究随机资产的风险值(VaR),证明了任意资产组合的风险不会超过其各个资产风险值之和,给出了组合资产风险值的上界。
风险结构问题是风险理论研究的重要内容,对金融企业控制风险有重要参考意义。本文研究了更加符合保险公司经营现实的风险模型及资产组合风险的VaR,对风险结构及相关问题进行探讨,着重从理论上研究如何预防破产的发生,所获得研究结果为金融企业控制风险提供理论依据。
Risk theory is a hot spot in modern actuarial science and risk structure is an important content in risk theory.On the assumption that risk is mutually independent, Classical risk theories give priority to stochastic risk models of insurance and survival or ruin probability within a limited time.In stochastic risk models,we consider mostly continual time risk models,especially compound Poisson procession risk models, discrete risk models are seldom treated,but discrete risk models are easier to perceive and more applicable.When treating discrete risk models,we often mention a compound binomial risk model whose characteristic is that the mean of claim numbers is greater than the variance of claim numbers,a compound binomial risk model is suitable for homogeneous policies assemblages,has been discussed.Actually policies assemblages have more or less non-homogeneity,a compound negative binomial risk model is more suitable for non-homogeneous policies assemblages than a compound binomial risk model and the property is that the variance of claim numbers is greater than the mean of claim numbers.This thesis Studies a compound negative binomial risk model whose formula of ultimate ruin probability has been proved by applying discrete martingale theory,the method is different form others references.We construct a compound negative binomial risk model with a completely stochastic premium where the premium of every policy and the number of insure charges at per unit time are random variables.This paper discusses some properties of a compound negative binomial risk model with a completely stochastic premium.By applying discrete martingale theory,this article proves the formula of ultimate ruin probability and the Lundberg inequality.
When only considering a policy whose premium and claim are independent each other,we can apply all sorts of risk models.But every insurer has a lot of policies whose premium and claim don't accord with existing risk models,this results that we can not apply the classics risk theories and use value at risk(VaR)to study a portfolio. Insures reinsure a policy whose loss exceeds the insurer's anticipation.Some scholars have studied an asset's VaR and stop-loss premium and some special portfolio's VaR, this thesis has discussed arbitrary asset portfolio's VaR and stop-loss premium, analytical expressions for risk value and stop-loss premiums of sums of independent random variables have been given.At finial,function convex order of non—negative random variables is introduced for random assets' VaR,the conclusion is that portfolio assets VaR can not exceed the sum of individual asset VaR,and the upper of portfolio is also given.
Risk structure problem is an important aspect of risk theory and is important significance for controlling risk of financial enterprises.This thesis has studied a new risk model which is more in line with the reality of insurers and arbitrary assets' portfolio's VaR,we study risk structure problem to guard against the occurrence of bankruptcy and our results are the gist for controlling risk of financial enterprises.
引文
[1] 熊福生 风险理论 . 武汉：武汉大学出版社,2005
[2] J.Dhaene,M.Denuit,M.J.Goovaerts,R.Kaas,R.Vyncke D. The cconcept of comonotonicity in actuarial science and finance :theory. Insurance:Mathematics and Economics,2002,(31):3-33
[3] J.Dhaene,M.Denuit,M.J.Goovaerts,R.Kaas,R.Vyncke D. The cconcept of in actuarial science and finance :applications. Insurance:Mathematics and Economics.2002,(31):133-161
[4] Kaas R., Van Heerwaarden,A.E.,Goovaerts. Ordering of actuarial risks. Caire Education Series ,1994
[5] H.De Meyera,B.De Baetsb,B. De Schuymera. On the transitivity of the comonotonic and countermonotonic comparison of random variables. Journal of Multivariate Analysis,2007,(98): 177-193
[6] Marc J.Goovaerts, Rob Kaas,Jan Dhaene,Qihe Tang. Some new classes of consistent risksk measures. Insurance:Mathematics and Economics, 2004, (34):505-516
[7] Inge Koch,Ann De Schepper. An application of comonotonicity and convex ordering to present values with truncated stochastic interest rates. Insurance Mathematics and Economics,2006
[8] Bounds on the value-at-risk for the sum of possibly dependent risks. Insurance:Mathematics and Economics,2005,(37):135-151
[9] Michele Vanmaele,Griselda Deelstra,Jan Liinev. Approximation of stop-loss premiums involving sums of lognormals by conditioning on two variables Insurance: Mathematics and Economics,2004,(35):343-367
[10] Michel Denuit,Jan Dhaene. Comonotonic bounds on the survival probabilities in the Lee-Carter model for mortality projection. Journal of Computational and Applied Mathematics
[11] Alessandro Juri, Mario V. Wuthrich. Copula convergence theorems for tail events. Insurance: Mathematics and Economics,2002,(30):405-420
[12] Jingping Yang,Shihong Cheng,Lihong Zhang. Bivariate copula decomposition in terms of comonotonicity,countermonotonicity and independence. Insurance: Mathematics and Economics,2006(39):267-284
[13] F.E. De Vylder,M.J. Goovaerts. Inequality extensions of Prabhu's formula in ruin theory. Insurance:Mathematics and Economics, 1999(24)249-271
[14] Scott Ferson,Janos G. Hajago. Varying correlation coefficients can underestimate uncertainty in probabilistic models. Reliability Engineering and System Safety, 2006(91):1461-1467
[15] Nicole Bauerle, Alfred Muller. Stochastic orders and risk measures:Consistency and bounds.Insurance:Mathematics and Economics,2006(38),132-148
[16]Goovaerts M.J.,Kaas R.,Van,Heerwaarden A.E.,et al.Effective actuarial methods.Amstrdam:North-Holland,1990
[17]Gerber H U.数学风险论导引.成世学,严颖译.北京:世界图书出版公司,1997
[18]严颖,成世学,程侃.运筹学随机模型.北京:中国人民大学出版社,1995
[19]Lundberg F.I,Approximerad Framstallning av Sannolikhetsfunktionen.Ⅱ,Atersforsakring av Kollektivrisker.Uppsala:Almqvist & Wiksell,1903
[20]Cramér H.On the Mathematical Theory of Risk.Stoockholm:Skandia Jubilee Volume,1930
[21]Cramér H.Collective Risk Theory.Stockholm:Skandia Jubilee Volume,1955
[22]Elliot R J.Stochastic Calculus and Applications.New York:Springer Verlag,1982
[23]Gerber H U.Martingale in risk theory.Mitt.Ver.Schweiz.Vers.Math.1973,73:205-216
[24]Bewere N L,et al.风险理论(中译本).上海:上海科学技术出版社,1995
[25]Bowers,N.L.,et al.风险理论.郑炜瑜,于跃年译.上海:上海科学技术出版1997
[26]Grandell J.Aspects of Risk Theory.New York:Spring verlag,1999:42-32
[27]乔克林,李晋枝,何树红.复合二项风险模型的破产概率.云南大学学报:自然科学版,2002,24(5):328-330
[28]龚日朝,杨向群.复合二项风险模型的破产概率.经济数学,2001,18(2):38-42
[29]Elliot R.J.Stochastic Calculus and Applications.New York:Springer Verlag,1982
[30]成世学.破产论研究综述.数学进展,2002,31(5):403-422
[31]严士健,刘秀芳.测度与概率论.北京:北京师范大学出版社,1994,54-137
[32]张茂军,南江霞.保费随机的复合二项风险模型的破产概率.科技通报,2005,21(3):367-371
[33]陈贵磊,赵明清.保费收取次数为负二项随机序列的复合二项风险模型.山东科技大学学报,2006,25(1):85-86
[34]高明美,赵明清.复合负二项风险模型的破产概率.山东科技大学学报(自然科学版),2006,18(1):102-104
[35]赵朋,马松林.双负二项风险模型的破产概率.合肥学院学报(自然科学版),2007,17(1):21-23
[36]R.卡尔斯,M.胡法兹,J.达呐,M.狄尼特.现代精算风险理论.唐启鹤,胡太忠,成世学译,北京:科学出版社,2005
[37]李维德,杨兵.证券投资组合的 VaR 模型风险分析.甘肃科学学报,2001,13(4):45-58
[38]毛泽春,刘锦萼.随机凸序与投资组合的风险值.经济数学学报,2004,21(1):16-23
[39]刘心声.非负随机变量的函数的凸序.安庆师院学报,1996,2(1):4-7
[40]刘心声.关于条件期望的随机序和凸序.厦门大学学报,1998,37(3):336-339
[41]贺思辉,李家军.相依随机风险变量下的金融风险测定-同单调随机序.统计理论与方法,2004,19(3):16-20
[42]金珩,王黎明.随机序关系及其在保险中的应用.内蒙古师范大学学报,2002,31(4):328-332
[43]龚海林,张新生.关于指数型分布族的随机序.华东师范大学学报,2004,3:48-53
[44]邱国新,冯胜章.齐次Possion冲击模型中元件寿命的随机比较.甘肃科学学报,2007,19(1):47-50
[45]陈贵磊.负合复合负二项风险模型研究:[山东科技大学硕士学位论文].青岛:山东科技大学,2005,6-15
[46]杨亚松.同单调风险及相关问题研究:[华东师范大学硕士学位论文].上海:华东师范大学,2004,3-24
[2] J.Dhaene,M.Denuit,M.J.Goovaerts,R.Kaas,R.Vyncke D. The cconcept of comonotonicity in actuarial science and finance :theory. Insurance:Mathematics and Economics,2002,(31):3-33
[3] J.Dhaene,M.Denuit,M.J.Goovaerts,R.Kaas,R.Vyncke D. The cconcept of in actuarial science and finance :applications. Insurance:Mathematics and Economics.2002,(31):133-161
[4] Kaas R., Van Heerwaarden,A.E.,Goovaerts. Ordering of actuarial risks. Caire Education Series ,1994
[5] H.De Meyera,B.De Baetsb,B. De Schuymera. On the transitivity of the comonotonic and countermonotonic comparison of random variables. Journal of Multivariate Analysis,2007,(98): 177-193
[6] Marc J.Goovaerts, Rob Kaas,Jan Dhaene,Qihe Tang. Some new classes of consistent risksk measures. Insurance:Mathematics and Economics, 2004, (34):505-516
[7] Inge Koch,Ann De Schepper. An application of comonotonicity and convex ordering to present values with truncated stochastic interest rates. Insurance Mathematics and Economics,2006
[8] Bounds on the value-at-risk for the sum of possibly dependent risks. Insurance:Mathematics and Economics,2005,(37):135-151
[9] Michele Vanmaele,Griselda Deelstra,Jan Liinev. Approximation of stop-loss premiums involving sums of lognormals by conditioning on two variables Insurance: Mathematics and Economics,2004,(35):343-367
[10] Michel Denuit,Jan Dhaene. Comonotonic bounds on the survival probabilities in the Lee-Carter model for mortality projection. Journal of Computational and Applied Mathematics
[11] Alessandro Juri, Mario V. Wuthrich. Copula convergence theorems for tail events. Insurance: Mathematics and Economics,2002,(30):405-420
[12] Jingping Yang,Shihong Cheng,Lihong Zhang. Bivariate copula decomposition in terms of comonotonicity,countermonotonicity and independence. Insurance: Mathematics and Economics,2006(39):267-284
[13] F.E. De Vylder,M.J. Goovaerts. Inequality extensions of Prabhu's formula in ruin theory. Insurance:Mathematics and Economics, 1999(24)249-271
[14] Scott Ferson,Janos G. Hajago. Varying correlation coefficients can underestimate uncertainty in probabilistic models. Reliability Engineering and System Safety, 2006(91):1461-1467
[15] Nicole Bauerle, Alfred Muller. Stochastic orders and risk measures:Consistency and bounds.Insurance:Mathematics and Economics,2006(38),132-148
[16]Goovaerts M.J.,Kaas R.,Van,Heerwaarden A.E.,et al.Effective actuarial methods.Amstrdam:North-Holland,1990
[17]Gerber H U.数学风险论导引.成世学,严颖译.北京:世界图书出版公司,1997
[18]严颖,成世学,程侃.运筹学随机模型.北京:中国人民大学出版社,1995
[19]Lundberg F.I,Approximerad Framstallning av Sannolikhetsfunktionen.Ⅱ,Atersforsakring av Kollektivrisker.Uppsala:Almqvist & Wiksell,1903
[20]Cramér H.On the Mathematical Theory of Risk.Stoockholm:Skandia Jubilee Volume,1930
[21]Cramér H.Collective Risk Theory.Stockholm:Skandia Jubilee Volume,1955
[22]Elliot R J.Stochastic Calculus and Applications.New York:Springer Verlag,1982
[23]Gerber H U.Martingale in risk theory.Mitt.Ver.Schweiz.Vers.Math.1973,73:205-216
[24]Bewere N L,et al.风险理论(中译本).上海:上海科学技术出版社,1995
[25]Bowers,N.L.,et al.风险理论.郑炜瑜,于跃年译.上海:上海科学技术出版1997
[26]Grandell J.Aspects of Risk Theory.New York:Spring verlag,1999:42-32
[27]乔克林,李晋枝,何树红.复合二项风险模型的破产概率.云南大学学报:自然科学版,2002,24(5):328-330
[28]龚日朝,杨向群.复合二项风险模型的破产概率.经济数学,2001,18(2):38-42
[29]Elliot R.J.Stochastic Calculus and Applications.New York:Springer Verlag,1982
[30]成世学.破产论研究综述.数学进展,2002,31(5):403-422
[31]严士健,刘秀芳.测度与概率论.北京:北京师范大学出版社,1994,54-137
[32]张茂军,南江霞.保费随机的复合二项风险模型的破产概率.科技通报,2005,21(3):367-371
[33]陈贵磊,赵明清.保费收取次数为负二项随机序列的复合二项风险模型.山东科技大学学报,2006,25(1):85-86
[34]高明美,赵明清.复合负二项风险模型的破产概率.山东科技大学学报(自然科学版),2006,18(1):102-104
[35]赵朋,马松林.双负二项风险模型的破产概率.合肥学院学报(自然科学版),2007,17(1):21-23
[36]R.卡尔斯,M.胡法兹,J.达呐,M.狄尼特.现代精算风险理论.唐启鹤,胡太忠,成世学译,北京:科学出版社,2005
[37]李维德,杨兵.证券投资组合的 VaR 模型风险分析.甘肃科学学报,2001,13(4):45-58
[38]毛泽春,刘锦萼.随机凸序与投资组合的风险值.经济数学学报,2004,21(1):16-23
[39]刘心声.非负随机变量的函数的凸序.安庆师院学报,1996,2(1):4-7
[40]刘心声.关于条件期望的随机序和凸序.厦门大学学报,1998,37(3):336-339
[41]贺思辉,李家军.相依随机风险变量下的金融风险测定-同单调随机序.统计理论与方法,2004,19(3):16-20
[42]金珩,王黎明.随机序关系及其在保险中的应用.内蒙古师范大学学报,2002,31(4):328-332
[43]龚海林,张新生.关于指数型分布族的随机序.华东师范大学学报,2004,3:48-53
[44]邱国新,冯胜章.齐次Possion冲击模型中元件寿命的随机比较.甘肃科学学报,2007,19(1):47-50
[45]陈贵磊.负合复合负二项风险模型研究:[山东科技大学硕士学位论文].青岛:山东科技大学,2005,6-15
[46]杨亚松.同单调风险及相关问题研究:[华东师范大学硕士学位论文].上海:华东师范大学,2004,3-24